The Negative Binomial Distribution

The negative binomial distribution is used when the number of successes is fixed and we're interested in the number of failures before reaching the fixed number of successes. An experiment which follows a negative binomial distribution will satisfy the following requirements:

1. The experiment consists of a sequence of independent trials.
2. Each trial has two possible outcomes, S or F.
3. The probability of success, , is constant from one trial to another.
4. The experiment continues until a total of r successes are observed, where r is fixed in advance.

A random variable X which follows a negative binomial distribution is denoted . Its probabilities are computed with the formula

for Formulas for E(X) and for the negative binomial distribution are given by

EXAMPLE:\ Suppose we are at a rifle range with an old gun that misfires 5 out of 6 times. Define ``success'' as the event the gun fires and let X be the number of failures before the third success. Then . The probability that there are 10 failures before the third success is given by

The expected value and variance of X are